Confusion of classification of closed surfaces

Question

I read that we can distinguish closed topological spaces without boundary up to homeomorphism by orientability and euler characteristic - is this correct?

But what confuses me is that the Klein bottle and the Mobius band are both non-orientable and both have Euler characteristic 0. But I thought theses spaces were not homeomorphic?

Please could you help me with this?

Answer 1

The Möbius band has a boundary, and therefore doesn't give a contradiction to the statement.